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Suppose \(x\) is a solution to \(x^2 + 1 = 7x\) . What is the sum of \(x\) and its reciprocal?

 Oct 22, 2019
 #1
avatar+976 
+1

You can simply find x and calculate

Eventually, this equals 7.

However, there is a faster way. Can you find out yourself?

 

You are very welcome!

:P

 Oct 22, 2019
 #2
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0

I think using the quadratic formula, we can find what x is equal to. 

so 

x^2+1=7x

Subtract 7x from both sides

x^2-7x+1

(Notice it is in the form ax^2+bx+c) 

Apply the quadratic formula

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

You will eventually get

x=6.85410196 approx.=6.85 

Now I don't understand what do you mean what is the sum of x? 

Its reciprocal is just 1/6.85 i guess.

 Oct 23, 2019
 #3
avatar+104969 
+1

Rearrange as

 

x^2  - 7x  = -1

 

Complete the square on x

 

x^2  - 7x  +  49/4  =  -1 + 49/4

 

(x - 7/2)^2  =  45/4       take the positive root

 

x - 7/2  =   √45 / 2

 

x - 7/2   =  3√5/2

 

x  =  [ 7 + 3√5]  / 2

 

Its reciprocal is    2  / [ 7 + 3√5] 

 

So

 

[ 7 + 3√5]  / 2  +   2  / [ 7 + 3√5]   =

 

 

[ 7 + 3√5] ^2  + 4

_______________   =

  2  [ 7 + 3√5] 

 

[ 49  + 42√5 + 45 + 4 ]

__________________   =

     2  [ 7 + 3√5] 

 

[  98 + 42 √5 ]

____________  =

  2  [ 7 + 3√5] 

 

 

49 + 21√5

_________  =

  7 +  3√5

 

[49 + 21√5] [ 7 - 3√5]

__________________   =

    49  - 45

 

 

343  + 147√5 - 147√5 - 63*5

_______________________  =

              4

 

343  -  315

_________   =

        4

 

28

__  =

 4

 

 

7

 

cool cool cool

 Oct 23, 2019

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